Vortex Identification Method Based on Topological Analysis and Velocity Gradient Invariance

Despite numerous works over the past two decades, friction-induced vibrations, especially braking noises, are a major issue for transportation manufacturers as well as for the scientific community. To study these fugitive phenomena, the engineers need numerical methods to efficiently predict the mode coupling instabilities in a multiparametric context. The objective of this paper is to approximate the unstable frequencies and the associated damping rates extracted from a complex eigenvalue analysis under variability. To achieve this, a deep Gaussian process is considered to fit the non-linear and non-stationary evolutions of the real and imaginary parts of complex eigenvalues. The current challenge is to build an efficient surrogate modelling, considering a small training set. A discussion about the sample distribution density effect, the training set size and the kernel function choice is proposed. The results are compared to those of a Gaussian process and a deep neural network. A focus is made on several deceptive predictions of surrogate models, although the better settings were well chosen in theory. Finally, the deep Gaussian process is investigated in a multiparametric analysis to identify the best number of hidden layers and neurons, allowing a precise approximation of the behaviours of complex eigensolutions.

This paper deals with a theoretical instability analysis of a rotating flexible disk subjected to swirling fluid flow in a confined fluid, and an instability mechanism governing the unstable vibration where the dissipation effect of structural damping and fluid viscosity are disregarded. In the instability analysis, the basic equations of swirling fluid flow around the rotating disk are derived by integrating Navier-Stokes equations over the gap width between the rotating disk wall and the shroud wall. The structural vibration equation of the rotating flexible disk is based on the Kirchhoff-Love plate model neglecting structural damping. The equations of coupled fluid-structure motion take into account the moving boundary conditions with respect to both the rotating disk and the fluid flow. These equations were linearized for small deflections of the disk near the equilibrium state, and the solution of these equations is obtained by using the multimodal expansion approximation and applying Galerkin's method. Instability is defined as the condition of a negative imaginary part of complex eigenvalues in the characteristics equation. The instability mechanism governing the unstable vibration is sufficiently understood. It is found that the characteristics of the unstable vibration are dependent on the added mass parameter.

Finite amplitude oscillations of flanged laminas in viscous flows: Vortex–structure interactions for hydrodynamic damping control

Centrifugal pump operation noise is an avoidable common phenomenon in engineering practices, and its characteristics are closely influenced by the flow characteristics of the fluid. In this paper, a numerical simulation of centrifugal pump operation with different flow rates is carried out using the detached eddy simulation turbulence model and compared with the experimental results. Then the noise source and far-field radiated noise characteristics of the centrifugal pump are investigated using Lighthill’s analogy. The results show that the noise of centrifugal pump is closely related to the flow characteristics of impeller, especially the generation and development of vortex. The relationship between turbulent kinetic energy (TKE) and vortex is complex, involving the generation, development and shedding of vortex. The influence of flow rate variation on vortex structure in impeller is studied by vortex transport equation. It is found that the relative vortex stretching (RVS) term is the main driving force of vortex distribution. The distribution of the Coriolis force (CORF) term is mainly concentrated in the inlet and outlet areas of the flow channel, which is related to the rotational motion of the impeller and the dynamic characteristic of the fluid. The power spectral density inside the impeller is mainly concentrated in the low frequency region, indicating that the low frequency noise component is dominant in energy. The directivity curve of far-field noise pressure level is influenced by hydrodynamic effect, impeller rotation direction and pump design structure, and will have a certain deviation. The spectral curve analysis of velocity, vorticity, pressure and sound pressure level shows that these physical quantities have strong correlation on the spectrum, in which the low frequency component is large, and the high frequency component is complex and volatile.

On the stability of certain systems under resonance conditions

We present a discussion of embedded vortices in general Yang-Mills theories. The origin of a family structure of solutions is shown to be group theoretic in nature and a procedure for its determination is developed. Vortex stability can be quantified into three types: Abelian topological stability, non-Abelian topological stability, and dynamical stability; we relate these to the family structure of vortices, in particular discussing how Abelian topological and dynamical stability are related. The formalism generally encompasses embedded domain walls and embedded monopoles also.

: The IUTAM Symposium in Ithaca, New York was the third of a series devoted to new discoveries in nonlinear dynamics an chaos related to mechanics, with Stuttgart 1990, and London, 1993 the prior venues for this meeting. These three symposia were focused on non-fluid phenomena especially solid and structural mechanics and multibody machine systems. However, a fair number of papers were devoted to fluid elastic and aeroelastic nonlinear dynamics. Nonlinear fluid mechanics is usually dominated by the quadratic nonlinear acceleration in the Navier Stokes equation. However, in solid, structural and machine mechanics there exists a greater variety of nonlinearities such as hysteresis, delay, nonsmooth discontinuities such a impact, and higher order nonlinearities in elastic structures as in shells and buckled structures.

Unstable transient response of gyroscopic systems with stable eigenvalues

In this paper, various solutions of the stationary Navier-Stokes equations, which describe the planar flow of an incompressible liquid (or gas), are determined, i.e., solutions containing the components of the velocity of flow – the functions u, v and the created pressure – P. The paper contains three proven theorems, as well as various examples and particular examined cases. Applying Theorem 1, we can find various solutions, where the velocity components represent the imaginary and real parts of a differentiable function of a complex variable. Theorem 2 allows us to determine solutions, where the velocity components are expressed by the partial derivatives of the solutions of Laplace’s equation of a special form. It is to be mentioned that these theorems give us solutions that do not depend on the viscosity parameter λ. In theorem 3, an original method for obtaining a series of solutions of the Navier-Stokes equations is presented, in which the viscosity coefficient λ participates explicitly; these solutions cannot be obtained by applying Theorems 1 or 2. The paper contains a large number of particular cases examined and examples of exact determined solutions.

Complex eigenvalue analysis is widely used when the dynamic instability of the structure is in doubt due to friction forces, aerodynamic forces, control systems, or other effects. MSC/NASTRAN upper Hessenberg method and MATLAB eigenvalue solver produce fictitious nonzero real parts for real asymmetric matrix eigenvalue problems. For dynamic instability problems, since nonzero real parts of complex eigenvalues determine the unstable eigenvalues, the accuracy of real parts becomes crucial. The appropriate double shift QR or the QZ algorithms are applied to eliminate fictitious nonzero real parts and produce only authentic complex eigenvalues for real asymmetric matrix eigenvalue problems. Numerical examples are solved using the double shift QR and the QZ algorithms, and the results are compared with the results of MSC/NASTRAN upper Hessenberg method and MATLAB solvers.

On complex eigenvalues of compartmental models

Management of the discrete eigenvalues of the nonlinear Schrödinger equation using periodic dispersion perturbation

In inviscid, incompressible flows, the evolution of vorticity is exactly equivalent to that of an infinitesimal material line-element and, hence, vorticity can be traced forward or backward in time in a Lagrangian fashion. This elegant and powerful description is not possible in viscous flows due to the action of diffusion. Instead, a stochastic Lagrangian interpretation is required and was recently introduced, where the origin of vorticity at a point is traced back in time as an expectation over the contribution from stochastic trajectories. We herein introduce for the first time an Eulerian, adjoint-based approach to quantify the back-in-time origin of vorticity in viscous, incompressible flows. The adjoint variable encodes the advection, tilting and stretching of the earlier-in-time vorticity that ultimately leads to the target value. Precisely, the adjoint vorticity is the volume-density of the mean Lagrangian deformation of the earlier vorticity. The formulation can also account for the injection of vorticity into the domain at solid boundaries. We demonstrate the mathematical equivalence of the adjoint approach and the stochastic Lagrangian approach. We then provide an example from turbulent channel flow, where we analyse the origin of high streamwise wall-shear-stress events and relate them to Lighthill’s mechanism of stretching of near-wall vorticity.

An improved ALE and CBS-based finite element algorithm for analyzing flows around forced oscillating bodies

This paper continues the studies of localized vortex evolution in nonparallel viscous plane flows with constant velocity shear started in our previous paper on the hyperbolic flow. This paper discusses the elliptic flow. For such a flow the evolution of a disturbance in the form of a monochromatic plane wave with a time-dependent wave vector was first investigated by Bayly [Phys. Rev. Lett. 57, 2160 (1986)] and also by Landman and Saffman [Phys. Fluids 30, 2339 (1987)] based on the Floquet theory. Here we address the evolution of a three-dimensionally localized vortex which is a package of such waves and includes both stable and exponentially unstable components. The effect of ellipticity on the character of vortex evolution is studied, as well as the possibility of its intensity amplification by a given instant of time. The dynamics of the geometrical shape of the vortex is calculated. The research is motivated by an attempt to interpret experiments on artificial generation of hairpin vortices in shear flows.